1. Newton’s law of motion:

a) Newton’s three laws of motion:
⇒ Newton’s First Law of Motion (Law of Inertia)

• An object at rest will remain at rest, and an object in motion will continue to move with a constant velocity, unless acted upon by an external force.
• – Inertia is the tendency of an object to resist changes in its motion.
• Example:
• (1)
• A skydiver descends to the earth with a parachute at a steady speed.
• The skydiver and parachute are in contact with nothing here, hence the free body diagram is simple (save the air).
• The weight of the skydiver and parachute, W, is balanced downward by the drag, D, pointing above.
• 
  Figure 1 Skydiver falling to the ground at a constant speed.
• (2)
• A climber is shown abseiling down a rock face in Figure 2 (1);
• He has just stopped to rest and is motionless.
• We have to take off the rock face in order to construct a free body diagram.
• Figure 2(2) illustrates the three forces at work:
• – The climber’s weight (W)
• – The rope’s tension (T)
• – He rocks face’s response (R).
• These forces act via the climber’s centre of gravity since they are immobile.
• They sum up to zero, as Figure 2(3) illustrates.
  
• Figure 2 (1) A climber is abseiling down a rock face, (2) The forces acting on the climber, (3) The forces add up to zero.
•  
• (3)
• A guy is seen ascending a ladder that leans against a smooth wall in Figure 3 (1).
• A free body diagram of the ladder with the guy standing on it is shown in Figure 3 (2).
  
• Figure 3 (1) A man climbing a ladder (2) A free body diagram for the man on the ladder.
• The forces acting on the ladder are:
• – [math]R_W[/math], a horizontal reaction force from the wall
• – [math]R_F[/math], a vertical reaction force from the floor
• – F, a horizontal frictional force from the floor
• – W, the weight of the ladder
• – [math]R_m[/math], a contact force from the man that is equal in size to his weight (this is not the man’s weight, which acts on him).
• Since the ladder remains stationary, the forces on it balance. So
• [math]R_F = W + R_m [/math]
• These are the forces acting vertically
• [math]F = R_W[/math]
• These are the forces acting horizontally

⇒ Newton’s Second Law of Motion (Law of Acceleration)

• The force applied to an object is equal to the mass of the object multiplied by its acceleration.
• [math]F = ma ( \text{Force = mass x acceleration}) [/math]
• – The more massive an object is, the more force is required to produce a given acceleration.
• Examples:
• (1)

  
  Figure 4 The forces on a cyclist

• Figure 4 shows the forces on a cyclist accelerating along the road. The forces in the vertical direction balance, but the force pushing her along the road, F, is greater than the drag forces, D, acting on her. The mass of the cyclist and the bicycle is 100kg. Calculate her acceleration.
• Given Data:
• Drag Force = [math]\vec{D} = 180 N[/math]
• Road force = [math]\vec{F} = 320N[/math]
• The mass of the cyclist and the bicycle= 100kg
• Find data:
• Acceleration =?
• Formula:
• [math]\vec{F} – \vec{D} = m\vec{a}[/math]
• Solution:
• Her acceleration can be calculated as follows.
• [math]\vec{F} – \vec{D} = m\vec{a}[/math]
• Put values
• [math]320 -180 =100 * \vec{a}[/math]
• [math]\frac{140}{100} = \vec{a} [/math
• [math]\vec{a}=1.4m/s[/math]
• (2)
• According to Newton’s second rule of motion, an object’s acceleration is exactly proportional to the net force it encounters
• Force = mass * acceleration
• Changing any of those three variables will change an object’s motion. Any change in force is proportional to a change in mass and/or acceleration, and any change in mass is inversely proportional to a change in acceleration (and vice versa).
• Figure 5 By 2^nd law force on a ball applies with respect to its acceleration
• For example, if an object of mass m is replaced by an object with double the mass, 2m, you would need twice the force to move the larger mass at the same acceleration as the smaller mass. Or, of you wanted to apply the same force to both masses, the 2m mass would have to move at half the acceleration as m.
• Because both acceleration and force are vector quantities (meaning they have a specific direction) changing the direction of a force will also result in a change in the direction of acceleration.

• For every action, there is an equal and opposite reaction.
• – When object A exerts a force on object B, object B will exert an equal and opposite force on object A.
• – This law describes the interaction between two objects, and how they respond to each other’s forces.
• Examples:
  (1)
• When the wheel of a car turns, it pushes the road backwards. The road pushes the wheel forwards with an equal and opposite force.
• Figure 6 Wheel of a car
• (2)
• 
• Figure 7 Two persons apply equal forces but in opposite direction
• (3)
• 
• Figure 8 If I push you with a force of 100N, you push me back with a force of 100N.
• (4)
• Two balloons have been charged positively. They each experience a repulsive force from the other.
• These forces are of the same size, so each balloon (if of the same mass) is lifted through the same angle.
• 
  Figure 9 Two balloons have been same charged (positively)

b) Momentum

• Momentum is a measure of an object’s mass and velocity.
• It’s a vector quantity, meaning it has both magnitude (amount of movement) and direction.
• The momentum of an object is calculated as:
• [math] \text{Momentum} \, (\vec{p}) = \text{Mass} \, (m) \times \text{Velocity} \, (\vec{v}) \\ \vec{p} = m \vec{v} [/math]
• The unit of momentum is typically measured in kilogram-meters per second (kg · m/s).
• Figure 10 Momentum is transferred from one object to another
• Here are some key aspects of momentum in physics:
• – Newton’s laws: Momentum is closely related to Newton’s laws of motion.
• -The second law states that force ([math]\vec{F}[/math]) is equal to the rate of change of momentum ([math] \Delta \vec{p} / \Delta t [/math]).
• – Momentum transfer: When objects interact, momentum is transferred from one object to another. This is why a baseball bat can transfer momentum to a ball, making it fly!
• – Relativity: In special relativity, momentum is affected by an object’s speed and mass. As an object approaches the speed of light, its momentum increases exponentially.
• – In the example of the exploding firework, chemical potential energy is transferred to thermal energy, light energy and kinetic energy of the exploding fragments.
• However, during the explosion the momentum remains the same.

• Example

• (1)
• A ball of mass 0.1kg hits the ground with a velocity of 6m/s and sticks to the ground. Calculate its change of momentum.
  Given data:
  Mass of a ball = m = 0.1 kg
  Change in velocity = [math]\Delta \vec{v}[/math] = 6m/s
  Find data:
  Change in momentum=[math]\Delta \vec{p}[/math]  = ?
  Formula:

  [math] \Delta \vec{p} = m \Delta \vec{v} [/math]

  Solution:

  [math] \Delta \vec{p} = m \Delta \vec{v} [/math]

  Put values

  [math] \Delta \vec{p} = 0.1(6) [/math]

  [math] \Delta \vec{p} = 0.6 kg .\frac{m}{s} [/math]

c) Momentum and impulse

• Momentum and impulse are closely related concepts in physics.
• Impulse is a measure of the change in momentum of an object.
• In other words, impulse is the product of the force applied to an object and the time over which it’s applied.
• The unit of impulse is typically measured in newton-seconds (N·s).
• Here are some key points about the relationship between momentum and impulse:
• – Impulse changes momentum: When an impulse is applied to an object, it changes the object’s momentum.
• – Momentum conservation: If the total impulse is zero, the total momentum remains constant.
• – Force and time tradeoff: A smaller force applied over a longer time can produce the same impulse as a larger force applied over a shorter time.
• – Impulse- momentum theorem: The impulse applied to an object is equal to the change in momentum of the object.
• Newton’s second law can be used to link an applied force to a change of momentum:
• [math] \vec{F} = m \vec{a} [/math]
• Substituting
• [math] \vec{a} = \frac{\Delta \vec{v}}{\Delta t} [/math]
• Gives
•  [math] \vec{F} = \frac{ m\Delta \vec{v}}{\Delta t} [/math]

• Or

  [math] \vec{F} = \frac{\Delta (m\vec{v})}{\Delta t} [/math]

• So,

• Force equals the rate of change of momentum.
• This is a more general statement of Newton’s second law of motion.
• The last equation may also be written in the form:
• [math] \vec{F} \Delta t = \Delta (m \vec{v}) [/math]
• or
• [math] \vec{F} \Delta t = m \vec{v}_2 – m \vec{v}_1 [/math]
• where [math]\vec{v_2}[/math] is the velocity after a force has been applied and [math]\vec{v_1}[/math] the velocity before the force was applied.
• The quantity [math] \vec{F} \Delta t = I [/math] is called the impulse. Where [math] \vec{F} [/math]  is constant act.
• 
  Figure 11 Impulse and momentum

Figure 12 Force–time graphs for two passengers in a car crash.

4. Collision 

a) The principle of conservation of momentum:

• • Conservation of linear momentum states that in a closed system, the total linear momentum remains constant over time.
  • This means that the total momentum before an event (like a collision) is equal to the total momentum after the event.
  • Mathematically, this is expressed as:
    Before collusion (initial) total momentum ([math]P_i[/math]) = After collusion (final) total momentum ([math]P_f[/math])
  • [math]P_i = P_f \\ m\vec{v-i} = m\vec{v-f}[/math]
  • For two bodies
  • [math] m_1 \vec{v}_1 + m_2 \vec{v}_2 = m_1 \vec{v}’_1 + m_2 \vec{v}’_2 [/math]
  • The total momentum of two bodies in a collision (or explosion) is the same after the collision (or explosion) as it was before.
  • 
    Figure 13 Law of conservation of momentum
  • This law applies to all closed systems, where no external forces are acting. In other words, the total momentum is conserved if there are no external influences that can change the momentum.
  • Momentum is always conserved in bodies involved in collisions and explosions provided no external forces act. This is the principle of conservation of momentum.
  • Here are some key aspects of conservation of linear momentum:
  • – Closed system: The system must be closed, meaning no objects can enter or leave, and no external forces can act upon it.
  • – Total momentum: The total momentum is the vector sum of all individual momenta in the system.
  • – Conservation: The total momentum remains constant over time, unless acted upon by an external force.
  • – Collisions: In collisions, momentum is transferred between objects, but the total momentum remains conserved.
  • – Explosions: In explosions, momentum is conserved, but the total momentum is distributed among the products.
  • Figure 13 shows a demonstration of a small one-dimensional explosion.
  • The head of a match has been wrapped tightly in aluminium foil.
  • A second match is used to heat the foil and the head of the first match, which ignites and explodes inside the foil.
  • The gases produced cause the foil to fly rapidly one way, and the matchstick to fly in the opposite direction. (It is necessary to use ‘strike-anywhere’ matches, the heads of which contain phosphorous.
  • The experiment is safe because the match blows out as it flies through the air, but common-sense safety precautions should be taken – do this in the laboratory, not in a carpeted room at home, and dispose of the matches afterwards.)
  • 
    Figure 14 A simple experiment that demonstrates the conservation of linear momentum
  • In all collisions and explosions, both total energy and momentum are conserved, but kinetic energy is not always conserved.
  • In this case, the chemical potential energy in the match head is transferred to the kinetic energy of the foil and matchstick, and also into thermal, light and sound energy.
  • As the match head explodes, Newton’s third law of motion tells us that both the matchstick and foil experience equal and opposite forces, [math] \vec{F} [/math].
  • Since the forces act for the same interval of time, Δt, both the matchstick and foil experience equal and opposite impulses, [math] \vec{F} \Delta t[/math].
  • Since [math] \vec{F} \Delta t = \Delta(m\vec{v})[/math], it follows that the foil gains exactly the same positive momentum as the matchstick gain’s negative momentum.
  • We can now do a vector sum to find the total momentum after the explosion (see Figure 5c):
  • [math] + \Delta (m \vec{v}) + \left[ – \Delta (m \vec{v}) \right] = 0[/math]
  • So, there is conservation of momentum: the total momentum of the foil and matchstick was zero before the explosion, and the combined momentum of the foil and matchstick is zero after the explosion.

  • ⇒ Collisions on an air track

  • Collision experiments can be carried out using gliders on linear air tracks.
  • These can demonstrate the conservation of linear momentum.
  • The air blowing out of small holes in the track lifts the gliders so that frictional forces are very small.

Figure 15 A laboratory air track with gliders

• ⇒Perfectly elastic collision and inelastic collision.
• Perfectly Elastic Collision:
• – A collision where both momentum and kinetic energy are conserved
• – The objects bounce off each other, retaining their original shape and size
• – The coefficient of restitution (e) is equal to 1 (e = 1)
• Examples:
• – Billiard balls
• – Superballs
• – Atomic or subatomic particle collisions
• Inelastic Collision:
• – A collision where momentum is conserved, but kinetic energy is not
• – The objects deform or stick together, losing some kinetic energy
• – The coefficient of restitution (e) is less than 1 (e < 1)
• Examples:
• – Car crashes
• – Football tackles
• – Dropping a ball on the ground
• In inelastic collisions
•  Some of the kinetic energy is converted into other forms, such as:
• – Heat
• – Sound
• – Deformation energy